Degree formula for the discriminant divisor of lagrangian fibrations of irreducible symplectic manifolds
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Abstract
Let \(M\) be an irreducible holomorphic symplectic manifold with a Lagrangian
fibration \(f: M \to \mathbb P^n\), whose discriminant locus is
\(\Delta = \bigcup_i \Delta_i \subset \mathbb P^n\). This thesis defines weights
\(w_i \in \mathbb Q\) such that
\[
24 \Biggl(\frac{n! \int_M \sqrt{\hat A}(M)}{d_1 \dotsm
d_n}\Biggr)^{\frac 1 n}
= \sum_i w_i \deg(\Delta_i),
\]
where \((d_1, \dotsc, d_n)\) is the polarization type of \(f\).
The definition of the \(w_i\) involves the cohomology sheaves of the
\(\Omega T\) complex
\[
f^* \Omega_{\mathbb P^n} \to \Omega_M \cong T_M \to f^* T_{\mathbb P^n},
\]
and this thesis gives an in-depth analysis of those sheaves.
Furthermore, the definition involves the choice of a Kähler form \(\omega\)
on \(M\), which induces a polarization of type \((d_1, \dotsc, d_n)\) on the
smooth fibers of \(f\). To show that the \(w_i\) do not depend on the choice
of \(\omega\) is the main undertaking of this thesis.
If the characteristic cycle \(\Theta_i\) over \(\Delta_i\) is compact, then
one can define the weights as
\[
w_i = \frac{\chi(\Theta_i)}{\deg_{\Theta_i}(\omega)}.
\]
In case of non-compact characteristic cycles one can choose an appropriate
compact subcycle \(\overline{\Theta}_i \subset \Theta_i\) to compute \(w_i\) in
the same way.